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IEEE Transactions on Quantum Engineering

Improved BP Decoding With Multiple Error-Path Detection for Topological Quantum Codes

Jaemin Kim, Hyunwoo Jung, Jeongseok Ha

Published Jan 1, 2026• 0 citations

Abstract

Topological quantum codes are promising candidates for realizing fault-tolerant quantum computing. In this context, belief propagation (BP) decoding is an efficient method for topological codes, offering high error-correction performance with low complexity. However, BP decoding suffers significant performance degradation due to quantum trapping sets (QTSs). Although various decoding schemes have been proposed to mitigate the performance degradation caused by QTSs, they incur high computational complexity and/or long latency. Moreover, they do not effectively address the presence of multiple QTSs that frequently arise as independent local structures in topological codes. To overcome this technical challenge, we propose a novel decoding scheme consisting of BP decoding and a low-complexity error-path detector. In particular, the proposed decoding scheme is designed to effectively break QTSs by exploiting paths with erroneous variable nodes (VNs) between unsatisfied check nodes. The proposed error-path detector identifies a set of paths associated with QTSs present in BP decoding failures. Then, the QTS-induced decoding failures are resolved by re-executing the BP decoding after modifying the prior information for the erroneous VNs in the identified paths. Numerical results show that the proposed decoding scheme achieves superior error-correction performance over state-of-the-art decoding schemes for various topological codes. Furthermore, we show that the proposed error-path detector operates with negligible computational overhead compared to BP per iteration. Thus, the proposed decoding scheme maintains the computational efficiency of BP decoding while effectively breaking multiple QTSs, making it a practical solution for decoding topological codes.

DOI: 10.1109/tqe.2026.3740093

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Topological quantum codes are promising candidates for realizing fault-tolerant quantum computing. In this context, belief propagation (BP) decoding is an efficient method for topological codes, offering high error-correction performance with low complexity. However, BP decoding suffers significant performance degradation due to quantum trapping sets (QTSs). Although various decoding schemes have been proposed to mitigate the performance degradation caused by QTSs, they incur high computational complexity and/or long latency. Moreover, they do not effectively address the presence of multiple QTSs that frequently arise as independent local structures in topological codes. To overcome this technical challenge, we propose a novel decoding scheme consisting of BP decoding and a low-complexity error-path detector. In particular, the proposed decoding scheme is designed to effectively break QTSs by exploiting paths with erroneous variable nodes (VNs) between unsatisfied check nodes. The proposed error-path detector identifies a set of paths associated with QTSs present in BP decoding failures. Then, the QTS-induced decoding failures are resolved by re-executing the BP decoding after modifying the prior information for the erroneous VNs in the identified paths. Numerical results show that the proposed decoding scheme achieves superior error-correction performance over state-of-the-art decoding schemes for various topological codes. Furthermore, we show that the proposed error-path detector operates with negligible computational overhead compared to BP per iteration. Thus, the proposed decoding scheme maintains the computational efficiency of BP decoding while effectively breaking multiple QTSs, making it a practical solution for decoding topological codes.

Quantum error correction (QEC) is critical for scalable fault-tolerant quantum computing. Topological codes, such as the toric code, offer hardware-efficient architectures but their Tanner graphs contain many girth-4 cycles that degrade the performance of belief-propagation (BP) decoding. For this reason, BP decoding is typically followed by a more complex second stage decoder such as minimum-weight perfect matching. These combined decoders achieve a remarkable performance, albeit at the cost of increased complexity. In this paper we propose two key improvements for the decoding of toric code. The first one is replacing the BP decoder by a neural BP decoder, giving rise to the neural belief-matching decoder which substantially decreases the average decoding complexity. The main drawback of this approach is the high cost associated with the training of the neural BP decoder. To address this issue, we impose a convolutional architecture on the neural BP decoder, enabling weight sharing across the spatially homogeneous structure of the code's factor graph. This design allows a model trained on a modest-size topological code to be directly transferred to much larger instances, preserving decoding quality while dramatically lowering the training burden. Our numerical experiments on toric-code lattices of various sizes demonstrate that this technique does not result in a noticeable loss in performance.

Topological codes, a kind of quantum error correction code, have been used for current quantum computers due to their local qubit layout and high threshold. With nearly linear complexity, syndrome-based belief propagation (BP) can be considered as a decoding candidate for topological codes. However, such highly degenerate codes will lead to multiple low-weight errors where the syndrome is identical so that the BP decoding is not able to distinguish it, resulting in degradation in performance. In this article, we propose a branch-assisted sign-flipping belief propagation (BSFBP) decoding method for topological codes based on the hypergraph product structure. In our algorithm, we introduce the criteria to enter the new decoding path branched from BP combined with a syndrome residual, which is obtained from the syndrome-pruning process. A sign-flipping process is also conducted to disturb the log-likelihood ratio of the selected variable nodes, which provides diversity in the syndrome residual. Simulation results show that using the proposed BSFBP decoding is able to outperform the BP decoding by about two orders of magnitude.

We propose Pathfinder-BP, a recursive decoder for topological quantum codes that combines belief propagation with controlled bit-flipping. An initial BP pass identifies and permanently fixes high-confidence errors. Upon convergence failure, the decoder recursively explores alternative error patterns by tentatively flipping low-confidence variables. On surface codes, Pathfinder-BP achieves a noise threshold and logical error rate comparable to minimum-weight perfect matching (MWPM). A crucial advantage of Pathfinder-BP is its highly parallelizable structure, which yields a worst-case decoding latency quasi-linear in the code distance, making it a suitable candidate for low-latency hardware implementation.

Quantum error correction is crucial for universal fault-tolerant quantum computing. Highly accurate and low-time-complexity decoding algorithms play an indispensable role in ensuring quantum error correction works effectively. Among existing decoding algorithms, belief propagation (BP) is notable for its nearly linear time complexity and general applicability to stabilizer codes. However, BP's decoding accuracy without post-processing is unsatisfactory in most situations. This article focuses on improving the decoding accuracy of BP over GF(4) for surface codes. Inspired by machine learning optimization techniques, we first propose Momentum-BP and AdaGrad-BP to reduce oscillations in message updating, breaking the trapping sets of surface codes. We further propose EWAInit-BP, which adaptively updates initial probabilities and provides a 1 to 3 orders of magnitude improvement over traditional BP for planar surface code, toric code, and XZZX surface code without any post-processing method, showing high decoding accuracy even under parallel scheduling. The theoretical

O(1)

time complexity under parallel implementation and high accuracy of EWAInit-BP make it a promising candidate for high-precision real-time decoders.

Quantum error-correcting codes (QECCs) are necessary for fault-tolerant quantum computation. Surface codes are a class of topological QECCs that have attracted significant attention due to their exceptional error-correcting capabilities and easy implementation. In the decoding process of surface codes, the syndromes are crucial for error correction, however, they are not always correctly measured. Most of the existing decoding algorithms for surface codes need extra measurements to correct syndromes with errors, which implies a potential increase in inference complexity and decoding latency. In this paper, we propose a high-performance list decoding algorithm for surface codes with erroneous syndromes, where syndrome soft information is incorporated in the decoding, allowing qubits and syndrome to be recovered without needing extra measurements. Precisely, we first use belief propagation (BP) decoding for pre-processing with syndrome soft information, followed by ordered statistics decoding (OSD) for post-processing to list and recover both qubits and syndromes. Numerical results demonstrate that our proposed algorithm efficiently recovers erroneous syndromes and significantly improves the decoding performance of surface codes with erroneous syndromes compared to minimum-weight perfect matching (MWPM), BP and original BP-OSD algorithms.

Iterative decoders for finite length quantum low-density parity-check (QLDPC) codes are attractive because their hardware complexity scales only linearly with the number of physical qubits. However, they are impacted by short cycles, detrimental graphical configurations known as trapping sets (TSs) present in a code graph as well as symmetric degeneracy of errors. These factors significantly degrade the decoder decoding probability performance and cause so-called error floor. In this paper, we establish a systematic methodology by which one can identify and classify quantum trapping sets (QTSs) according to their topological structure and decoder used. The conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. We show that the knowledge of QTSs can be used to design better QLDPC codes and decoders. Frame error rate improvements of two orders of magnitude in the error floor regime are demonstrated for some practical finite-length QLDPC codes without requiring any post-processing.